Why is my texture applying itself to unwanted objects?
Both the body and the hair mesh are using the same image (Untitled).

In BI the image displayed on the mesh in textured view is determined by what image is displayed in the UV image editor while that geometry is selected.

Create a new separate image for the hair (AltN in the UV/image editor), then "assign" the hair geometry to that image by selecting it in the image editor with all of the hair geometry selected in edit mode.

Note that this doesn't affect the final render unless you have Face textures enabled.

See How is this blendfile hiding textures? and Texture mapped incorrectly in render?